95-270 del Rio R., Jitomirskaya S., Last Y., Simon B.

Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization (112K, AMSTeX) Jun 14, 95

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Abstract. Although concrete operators with singular continuous spectrum have proliferated recently, we still don't really understand much about singular continuous spectrum. In part, this is because it is normally defined by what it isn't --- neither pure point nor absolutely continuous. An important point of view, going back in part to Rodgers and Taylor, and studied recently within spectral theory by Last, is the idea of using Hausdorff measures and dimensions to classify measures. Our main goal in this paper is to look at the singular spectrum produced by rank one perturbations from this point of view. A Borel measure $\mu$ is said to have exact dimension $\alpha\in [0,1]$ if and only if $\mu(S)=0$ if $S$ has dimension $\beta <\alpha$ and if $\mu$ is supported by a set of dimension $\alpha$. If $0<\alpha <1$, such a measure is, of necessity, singular continuous. But, there are also singular continuous measures of exact dimension $0$ and $1$ which are ``particularly close'' to point and a.c.~measures, respectively. Indeed, as we'll explain, we know of ``explicit'' Schr\"odinger operators with exact dimension $0$ and $1$, but, while they presumably exist, we don't know of any with dimension $\alpha\in (0,1)$. While we're interested in the abstract theory of rank one perturbations, we're especially interested in those rank one perturbations obtained by taking a random Jacobi matrix and making a Baire generic perturbation of the potential at a single point. It is a disturbing fact that the strict localization (dense point spectrum with $\|xe^{-itH}\delta_{0}\|^2 = (e^{-itH}\delta_{0},\, x^2 e^{-itH}\delta_{0})$ bounded in $t$), that holds a.e.~for the random case, can be destroyed by arbitrarily small local perturbations. We'll ameliorate this discovery in the present paper in three ways: First, we'll see that, in this case, the spectrum is always of dimension zero, albeit sometimes pure point and sometimes singular continuous. Second, we'll show that not only does the set of couplings with singular continuous spectrum has Lebesgue measure zero, it has Hausdorff dimension zero. Third, we'll also see that while $\|xe^{-itH} \delta_{0}\|$ may be unbounded after the local perturbation, it never grows faster than $C\ln(t)$. Appendix 2 contains an example of a Jacobi matrix which sheds light on the proper definition of localization: It has a complete set of exponentially decaying eigenfunctions, but, nevertheless, $\varlimsup\limits_{t\to\infty} \|xe^{itH}\delta_{0}\|^{2}/t^{\alpha} = \infty$ for any $\alpha <2$. Section 7 discusses further the connection between eigenfunction localization and transport. In Section 2, we'll review some basic facts about Hausdorff measures that we'll use later. In Section 3, we relate these to boundary behavior of Borel transforms. In Section 4, we use these ideas to present relations between spectra produced by rank one perturbations and the behavior of the spectral measure of the unperturbed operator. In Section 5, we'll relate Hausdorff dimensions of some energy sets to the dimensions of some coupling constant sets. In Section 6, we use the results of Sections 4 and 5 to present examples that show that the Hausdorff dimension under perturbation can be anything. In Section 7, we turn to systems with exponentially localized eigenfunctions, and show that under local perturbations the spectrum remains of Hausdorff dimension zero. Some of the lemmas in this section on the nature of localization are of independent interest. Finally, in Section 8, we prove that ``physical'' localization is ``almost stable,'' that is, suitable decay of $(\delta_{n}, e^{-itH}\delta_{m})$ in $|n-m|$ uniform in $t$ implies that $\|x\exp (-it(H+\lambda \delta_{0}))\delta_{0}\|$ grows at worst logarithmically. Appendix 1 provides a proof of a variant of a theorem of Aizenman relating Green's function estimates to dynamics and Appendix 2 is an example with interesting pathologies. Appendix 3 shows that our notion of ``semi-uniform'' localization introduced in Section 7 cannot be replaced by uniform localization for the Anderson model. Appendix 4 extends a lemma of Howland to allow consideration of dimension and Appendix 5 provides the technical details of one class of examples in Section 6.

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